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Physics 102: Lecture 25
Atomic Spectroscopy & Quantum Atoms
Physics 102: Lecture 25, Slide 1
PICK UP A DIFFRACTION GRATING!
From last lecture – Bohr model
Angular momentum is quantized
Ln = nh/2π
n = 1, 2, 3 ...
Energy is quantized
mk 2e4 Z 2
13.6  Z 2
En  

eV  where  h / 2 
2
2
2
2
n
n
Radius is quantized
2
2
n2
 h  1 n
rn  
  0.0529 nm 

2
Z
 2  mke Z
Velocity too!
Physics 102: Lecture 25, Slide 2
+Ze
Transitions + Energy Conservation
• Each orbit has a specific energy:
En= -13.6 Z2/n2
• Photon emitted when electron
jumps from high energy to low
energy orbit. Photon absorbed
when electron jumps from low
energy to high energy:
E2 – E1 = h f = h c / l
Physics 102: Lecture 25, Slide 3
E2
E1
Demo: Line Spectra
In addition to the continuous blackbody spectrum,
elements emit a discrete set of wavelengths which
show up as lines in a diffraction grating.
656 nm
H
n=3
n=2
This is how neon signs &
Na lamps work!
Spectra give us information on
atomic structure
Physics 102: Lecture 25, Slide 4
n=1
Solar spectrum!
Physics 102: Lecture 25, Slide 5
Checkpoint 1.1
Electron A falls from energy level n=2 to energy level n=1
(ground state), causing a photon to be emitted.
Electron B falls from energy level n=3 to energy level n=1
(ground state), causing a photon to be emitted.
n=3
n=2
Which photon has more energy?
1) Photon A
2) Photon B
A
B
n=1
Physics 102: Lecture 25, Slide 6
Spectral Line Wavelengths
Calculate the wavelength of photon emitted when an electron in the
hydrogen atom drops from the n=2 state to the ground state (n=1).
E2= -3.4 eV
n=3
n=2
Z2
E n  13.6eV 2
n
hf  E2  E1
 3.4eV  (13.6eV)  10.2eV
E1= -13.6 eV
Ephoton 
Physics 102: Lecture 25, Slide 7
n=1
hc
l
hc
1240
l

 124nm
10.2eV 10.2
ACT: Spectral Line Wavelengths
Compare the wavelength of a photon produced from a transition
from n=3 to n=2 with that of a photon produced from a transition
n=2 to n=1.
A
l32 < l21
B
l32 = l21
C
l32 > l21
E32 < E21
Physics 102: Lecture 25, Slide 8
so
n=3
n=2
l32 > l21
n=1
ACT/Checkpoint 1.2
The electrons in a large group of hydrogen atoms are
excited to the n=3 level. How many spectral lines will
be produced?
A. 1
B. 2
n=3
n=2
C. 3
D. 4
E. 5
n=1
Physics 102: Lecture 25, Slide 9
The Bohr Model is incorrect!
To be consistent with the Heisenberg Uncertainty Principle, which
of these properties cannot be quantized (have the exact value
known)? (A=OK
B= NOT )
Electron Radius
Would know location
Electron Energy
Electron Velocity
Would know momentum
Electron Angular Momentum
But, in the Bohr model:
2
2
2
h
1
n
n


rn  
  0.0529 nm 

2
Z
 2  mke Z
Physics 102: Lecture 25, Slide 10
Quantized radii
and velocities for
electron orbitals
Checkpoint 2
+Ze
Bohr Model
Quantum Atom
So what keeps the electron from “sticking” to the nucleus?
21%
36%
42%
A) Centripetal Acceleration
B) Pauli Exclusion Principle
C) Heisenberg Uncertainty Principle
Physics 102: Lecture 25, Slide 11
Quantum Mechanics
Theory used to predict probability distributions
QM
Physics 102: Lecture 25, Slide 12
Quantum Mechanical Atom
• Predicts available energy states agreeing with
Bohr.
• Don’t have definite electron position, only a
probability function.
• Each orbital can have 0 angular momentum!
• Each electron state labeled by 4 numbers:
n = principal quantum number (1, 2, 3, …)
l = angular momentum (0, 1, 2, … n-1)
ml = component of l (-l < ml < l)
ms = spin (-½ , +½)
Physics 102: Lecture 25, Slide 13
Quantum Mechanics (vs. Bohr)
Electrons are described by a probability function,
not a definite radius!
It takes 4 numbers to
describe the electron
Bohr: just 𝑛
Each orbital 𝑛 can
have 0 angular
momentum
Bohr: 𝐿𝑛 = 𝑛 ℏ
n = 1,2,3 …
Physics 102: Lecture 25, Slide 14
Quantum Numbers
Each electron in an atom is labeled by 4 #’s
n = Principal Quantum Number (1, 2, 3, …)
• Determines the Bohr energy
l = Orbital Quantum Number (0, 1, 2, … n-1)
•
•
Determines angular momentum
l <n
always true!
L
h
(  1)
2
ml = Magnetic Quantum Number (-l , … 0, … l )
•
•
z-component of l
| ml | <= l
always true!
ms = Spin Quantum Number (-½ , +½)
•
“Up Spin” or “Down Spin”
Physics 102: Lecture 25, Slide 15
h
Lz  m
2
ACT: Quantum numbers
For which state of hydrogen is the orbital
angular momentum required to be zero?
1. n=1
2. n=2
3. n=3
Physics 102: Lecture 25, Slide 16
The allowed values of l are
0, 1, 2, …, n-1. When n=1, l
must be zero.
Spectroscopic Nomenclature
“Shells”
“Subshells”
l =0 is “s state”
l =1 is “p state”
l =2 is “d state”
l =3 is “f state”
l =4 is “g state”
n=1 is “K shell”
n=2 is “L shell”
n=3 is “M shell”
n=4 is “N shell”
n=5 is “O shell”
1 electron in ground state of Hydrogen:
n=1, l =0 is denoted as: 1s1
n=1
Physics 102: Lecture 25, Slide 17
l =0
1 electron
Quantum Numbers
How many unique electron states exist with n=2?
l = 0 : 2s2
ml = 0 : ms = ½ , -½
2 states
l = 1 : 2p6
ml = +1: ms = ½ , -½
ml = 0: ms = ½ , -½
ml = -1: ms = ½ , -½
2 states
2 states
2 states
There are a total of 8 states with n=2
Physics 102: Lecture 25, Slide 18
Electron orbitals
In correct quantum mechanical description of atoms, positions of
electrons not quantized, orbitals represent probabilities
Carbon orbitals
imaged in 2009 using
electron microscopy!
Physics 102: Lecture 25, Slide 19
ACT: Quantum Numbers
How many unique electron states exist with n=5
and ml = +3?
A) 0
B) 4
C) 8
D) 16
E) 50
l
l
l
l
= 0 : ml = 0
= 1 : ml = -1, 0, +1
= 2 : ml = -2, -1, 0, +1, +2
Only
l = 3 and l = 4
have ml = +3
= 3 : ml = -3, -2, -1, 0, +1, +2, +3
ms = ½ , -½
2 states
l = 4 : ml = -4, -3, -2, -1, 0, +1, +2, +3, +4
ms = ½ , -½
2 states
There are a total of 4 states with n=5, ml = +3
Physics 102: Lecture 25, Slide 20
Pauli Exclusion Principle
In an atom with many electrons only one electron
is allowed in each quantum state (n, l, ml, ms).
This explains the periodic table!
Physics 102: Lecture 25, Slide 21
Electron Configurations
#
electrons Atom
Configuration
1
2
H
He
1s1
1s2
3
Li
1s22s1
4
Be
1s22s2
5
B
etc
1s22s22p1
10
Ne
(n=1 shell filled noble gas)
2s shell filled
1s22s22p6
s shells hold up to 2 electrons
Physics 102: Lecture 25, Slide 22
1s shell filled
2p shell filled
(n=2 shell filled noble gas)
p shells hold up to 6 electrons
The Periodic Table
s (l =0)
n = 1, 2, 3, ...
p (l =1)
Also s
d (l =2)
f (l =3)
What determines the sequence? Pauli exclusion & energies
Physics 102: Lecture 25, Slide 23
Summary
• Each electron state labeled by 4 numbers:
n = principal quantum number (1, 2, 3, …)
l = angular momentum (0, 1, 2, … n-1)
ml = component of l (-l < ml < l)
ms = spin (-½ , +½)
• Pauli Exclusion Principle explains periodic table
• Shells fill in order of lowest energy.
Physics 102: Lecture 25, Slide 24